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# The Mathematics of the Heavens and the Earth:The Early History of TrigonometryGlen Van Brummelen

 TABLE OF CONTENTS:Preface xiThe Ancient Heavens 1Chapter 1: Precursors 9What Is Trigonometry? 9The Seqed in Ancient Egypt 10* Text 1.1 Finding the Slope of a Pyramid 11Babylonian Astronomy, Arc Measurement, and the 360° Circle 12The Geometric Heavens: Spherics in Ancient Greece 18A Trigonometry of Small Angles? Aristarchus and Archimedes on Astronomical Dimensions 20* Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon 24Chapter 2: Alexandrian Greece 33Convergence 33Hipparchus 34A Model for the Motion of the Sun 37* Text 2.1 Deriving the Eccentricity of the Sun's Orbit 39Hipparchus's Chord Table 41The Emergence of Spherical Trigonometry 46Theodosius of Bithynia 49Menelaus of Alexandria 53The Foundations of Spherical Trigonometry: Book III of Menelaus's Spherics 56* Text 2.2 Menelaus, Demonstrating Menelaus's Theorem 57Spherical Trigonometry before Menelaus? 63Claudius Ptolemy 68Ptolemy's Chord Table 70Ptolemy's Theorem and the Chord Subtraction/Addition Formulas 74The Chord of 1° 76The Interpolation Table 77Chords in Geography: Gnomon Shadow Length Tables 77* Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths 78Spherical Astronomy in the Almagest 80Ptolemy on the Motion of the Sun 82* Text 2.4 Ptolemy, Determining the Solar Equation 84The Motions of the Planets 86Tabulating Astronomical Functions and the Science of Logistics 88Trigonometry in Ptolemy's Other Works 90* Text 2.5 Ptolemy, Constructing Latitude Arcs on a Map 91After Ptolemy 93Chapter 3: India 94Transmission from Babylon and Greece 94The First Sine Tables 95Aryabhata's Difference Method of Calculating Sines 99* Text 3.1 Aryabhata, Computing Sines 100Bhaskara I's Rational Approximation to the Sine 102Improving Sine Tables 105Other Trigonometric Identities 107* Text 3.2 Varahamihira, a Half-angle Formula 108* Text 3.3 Brahmagupta, the Law of Sines in Planetary Theory? 109Brahmagupta's Second-order Interpolation Scheme for Approximating Sines 111* Text 3.4 Brahmagupta, Interpolating Sines 111Taylor Series for Trigonometric Functions in Madhava's Kerala School 113Applying Sines and Cosines to Planetary Equations 121Spherical Astronomy 124* Text 3.5 Varahamihira, Finding the Right Ascension of a Point on the Ecliptic 125Using Iterative Schemes to Solve Astronomical Problems 129* Text 3.6 Paramesvara, Using Fixed-point Iteration to Compute Sines 131Conclusion 133Chapter 4: Islam 135Foreign Junkets: The Arrival of Astronomy from India 135Basic Plane Trigonometry 137Building a Better Sine Table 140* Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why the Circle Should Have 480 Degrees 146Introducing the Tangent and Other Trigonometric Functions 149* Text 4.2 Abu'l-Rayhan al-Biruni, Finding the Cardinal Points of the Compass 152Streamlining Astronomical Calculation 156* Text 4.3 Kushyar ibn Labban, Finding the Solar Equation 156Numerical Techniques: Approximation, Iteration, Interpolation 158* Text .4 Ibn Yunus, Interpolating Sine Values 164Early Spherical Astronomy: Graphical Methods and Analemmas 166* Text 4.5 Al-Khwarizmi, Determining the Ortive Amplitude Geometrically 168Menelaus in Islam 173* Text 4.6 Al-Kuhi, Finding Rising Times Using the Transversal Theorem 175Menelaus's Replacements 179Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of the Magnitudes and Nasir al-Din al-Tusi's Transversal Figure 186Applications to Religious Practice: The Qibla and Other Ritual Needs 192* Text 4.7 Al-Battani, a Simple Approximation to the Qibla 195Astronomical Timekeeping: Approximating the Time of Day Using the Height of the Sun 201New Functions from Old: Auxiliary Tables 205* Text 4.8 Al-Khalili, Using Auxiliary Tables to Find the Hour-angle 207Trigonometric and Astronomical Instruments 209* Text 4.9 Al-Sijzi (?), On an Application of the Sine Quadrant 213Trigonometry in Geography 215Trigonometry in al-Andalus 217Chapter 5: The West to 1550 223Transmission from the Arab World 223An Example of Transmission: Practical Geometry 224* Text 5.1 Hugh of St. Victor, Using an Astrolabe to Find the Height of an Object 225* Text 5.2 Finding the Time of Day from the Altitude of the Sun 227Consolidation and the Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard of Wallingford, and John of Murs 230* Text 5.3 Levi ben Gerson, The Best Step Size for a Sine Table 233* Text 5.4 Richard of Wallingford, Finding Sin(1°) with Arbitrary Accuracy 237Interlude: The Marteloio in Navigation 242* Text 5.5 Michael of Rhodes, a Navigational Problem from His Manual 244From Ptolemy to Triangles: John of Gmunden, Peurbach, Regiomontanus 247* Text 5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and Another Side 254* Text 5.7 Regiomontanus, the Angle-angle-angle Case of Solving Right Triangles 255Successors to Regiomontanus: Werner and Copernicus 264* Text 5.8 Copernicus, the Angle-angle-angle Case of Solving Triangles 267* Text 5.9 Copernicus, Determining the Solar Eccentricity 270Breaking the Circle: Rheticus, Otho, Pitiscus and the Opus Palatinum 273Concluding Remarks 284Bibliography 287Index 323Return to Book DescriptionFile created: 4/21/2017 Questions and comments to: webmaster@press.princeton.eduPrinceton University Press